Subgroups — #29698542

Order 15616.

808 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21454737	C₂	1	1	yes	maximal, characteristic
2	#21454735	C₂	1	1	yes	maximal, characteristic
3	C₂×#16839883	C₂	1	1	yes	maximal, characteristic
4	#21454429	C₂	1	2	yes	maximal, complemented
5	#21446610	C₂	1	1	yes	maximal, characteristic, complemented
6	#21446062	C₂	1	2	yes	maximal, complemented
7	#21446056	C₂	1	2	yes	maximal, complemented
8	#21445992	C₂	1	1	yes	maximal, characteristic, complemented
9	#21445872	C₂	1	2	yes	maximal, complemented
10	#21445704	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₁×#1953	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14519799	C₂²	1	1	yes	characteristic
13	#16840528	C₂²	1	2	yes	—
14	#16840528	C₂²	1	2	yes	—
15	C₂×#14519785	C₂²	1	2	yes	—
16	#16839885	—	2	2	no	—
17	#16839883	—	2	2	no	—
18	#16839883	C₄	1	4	yes	complemented
19	#16839883	C₄	1	4	yes	complemented
20	#16839880	—	2	2	no	—
21	#16839878	—	2	2	no	—
22	#16839811	C₂²	1	2	yes	—
23	#16839809	—	2	4	no	—
24	C₂×#14519713	C₂²	1	1	yes	characteristic
25	C₂×#14519712	C₂²	1	2	yes	—
26	C₂×#14519712	C₂²	1	1	yes	characteristic
27	#108×D₆₁	—	2	4	no	—
28	#16839798	C₂²	1	1	yes	characteristic
29	#16839785	C₂²	1	2	yes	—
30	#16839780	C₂²	1	2	yes	—
31	#16839778	C₂²	1	2	yes	—
32	C₂×#14519703	C₂²	1	2	yes	—
33	#16839717	C₂²	1	1	yes	characteristic
34	C₂×#14519687	C₂²	1	1	yes	characteristic
35	#16839534	C₂²	1	2	yes	—
36	#16839523	C₂²	1	1	yes	characteristic
37	#16839521	C₂²	1	1	yes	characteristic
38	C₆₁×#430	C₂²	1	2	yes	—
39	C₂×#108×C₆₁	C₂²	1	1	yes	characteristic
40	C₂×#108×C₆₁	C₂²	1	1	yes	characteristic
41	C₆₁×#405	C₂²	1	1	yes	characteristic
42	C₆₁×#344	C₂²	1	2	yes	—
43	C₂×#11758222	C₂³	1	2	yes	—
44	#14519799	C₂×C₄	1	2	yes	—
45	#14519799	C₂×C₄	1	2	yes	—
46	#14519799	D₄	1	2	yes	—
47	#14519799	D₄	1	2	yes	—
48	C₂×Dic₂₄₄	Q₈	1	1	yes	characteristic
49	C₂×#11758190	Q₈	1	2	yes	—
50	C₂×D₄₈₈	Q₈	1	1	yes	characteristic
51	#14519785	—	2	2	no	—
52	#14519785	—	2	2	no	—
53	#14519785	—	2	4	no	—
54	#14519785	—	4	2	no	—
55	#14519785	C₂×C₄	1	4	yes	—
56	#14519785	C₂×C₄	1	4	yes	—
57	#14519783	—	2	4	no	—
58	#14519783	—	4	2	no	—
59	C₂×#11758194	C₂³	1	2	yes	—
60	C₂×#11758194	C₂³	1	1	yes	characteristic
61	C₂×#11758189	—	2	2	no	—
62	M₁₆×D₆₁	—	2	4	no	—
63	M₁₆×D₆₁	—	2	4	no	—
64	C₂²×#92418	C₂³	1	1	yes	characteristic
65	C₂×C₈×D₆₁	—	2	2	no	—
66	#14519736	C₂³	1	1	yes	characteristic
67	C₂×C₄×Dic₆₁	C₂³	1	1	yes	characteristic
68	#14519713	—	2	1	no	—
69	#14519713	—	2	1	no	—
70	#14519713	—	2	4	no	—
71	#14519712	—	2	2	no	—
72	#14519712	—	2	2	no	—
73	#14519712	D₄	1	4	yes	—
74	#14519710	—	2	1	no	—
75	#14519708	—	2	1	no	—
76	#14519706	—	4	2	no	—
77	#14519705	—	2	4	no	—
78	#14519703	—	2	4	no	—
79	#14519703	C₂×C₄	1	8	yes	—
80	#14519687	—	2	1	no	—
81	#14519687	C₂×C₄	1	2	yes	—
82	#14519687	C₂×C₄	1	2	yes	—
83	C₈×Dic₆₁	—	2	1	no	—
84	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
85	C₂×M₁₆×C₆₁	C₂³	1	2	yes	—
86	C₂×M₁₆×C₆₁	C₂³	1	1	yes	characteristic
87	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
88	#117×C₆₁	C₂³	1	1	yes	characteristic
89	#108×C₆₁	—	2	4	no	—
90	#108×C₆₁	—	2	1	no	—
91	#108×C₆₁	—	2	1	no	—
92	#108×C₆₁	C₂×C₄	1	2	yes	—
93	#108×C₆₁	C₂×C₄	1	2	yes	—
94	#97×C₆₁	—	2	1	no	—
95	C₄×C₈×C₆₁	—	2	1	no	—
96	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
97	C₂×C₂²⋊D₆₁	—	2	2	no	—
98	C₂²×Dic₆₁	—	2	1	no	—
99	#11758222	—	2	2	no	—
100	#11758222	—	2	2	no	—
101	#11758222	C₂²×C₄	1	4	yes	—
102	#11758222	C₂×D₄	1	4	yes	—
103	C₂×D₂₄₄	C₂×Q₈	1	2	yes	—
104	C₂×C₄×D₆₁	—	2	2	no	—
105	C₂×Dic₁₂₂	C₂×Q₈	1	2	yes	—
106	M₁₆×C₆₁	—	2	2	no	—
107	M₁₆×C₆₁	—	2	1	no	—
108	M₁₆×C₆₁	—	2	2	no	—
109	M₁₆×C₆₁	—	2	2	no	—
110	M₁₆×C₆₁	C₂²×C₄	1	4	yes	—
111	M₁₆×C₆₁	C₂×D₄	1	2	yes	—
112	C₂×C₈×C₆₁	—	2	1	no	—
113	C₂×C₈×C₆₁	—	2	2	no	—
114	C₂×C₈×C₆₁	—	2	2	no	—
115	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
120	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
121	C₄⋊C₄×C₆₁	—	2	1	no	—
122	C₂²⋊C₄×C₆₁	—	2	1	no	—
123	C₄²×C₆₁	—	2	1	no	—
124	C₂²⋊Dic₆₁	—	2	1	no	—
125	C₄⋊Dic₆₁	—	2	1	no	—
126	C₄×Dic₆₁	—	2	1	no	—
127	C₄×Dic₆₁	—	2	1	no	—
128	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
129	#11758194	—	2	1	no	—
130	#11758194	—	2	2	no	—
131	#11758194	—	2	2	no	—
132	#11758194	C₂×D₄	1	2	yes	—
133	C₂×#92418	—	2	1	no	—
134	C₂×#92418	—	2	2	no	—
135	C₂×#92418	C₂²×C₄	1	2	yes	—
136	C₂×#92418	C₂×D₄	1	2	yes	—
137	C₂×#92418	#41	1	1	yes	characteristic
138	C₂×#92418	#41	1	1	yes	characteristic
139	Dic₂₄₄	C₄⋊C₄	1	2	yes	—
140	Dic₂₄₄	C₄⋊C₄	1	2	yes	—
141	D₄₈₈	C₄⋊C₄	1	2	yes	—
142	D₄₈₈	C₄⋊C₄	1	2	yes	—
143	#11758190	C₄⋊C₄	1	4	yes	—
144	#11758190	C₄⋊C₄	1	4	yes	—
145	#11758189	—	2	4	no	—
146	#11758189	—	2	4	no	—
147	#11758189	—	4	4	no	—
148	#11758189	—	4	4	no	—
149	C₈×D₆₁	—	2	4	no	—
150	C₈×D₆₁	—	2	4	no	—
151	C₈×D₆₁	—	4	4	no	—
152	C₈×D₆₁	—	4	4	no	—
153	C₂³×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
154	C₂²×D₆₁	—	2	2	no	—
155	C₂×C₄×C₆₁	—	4	1	no	—
156	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
157	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
160	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
161	C₂×C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
162	C₂²⋊D₆₁	—	2	4	no	—
163	C₂²⋊D₆₁	—	2	4	no	—
164	C₂×Dic₆₁	—	2	2	no	—
165	C₂×Dic₆₁	—	2	2	no	—
166	C₂×Dic₆₁	—	4	1	no	—
167	C₂×Dic₆₁	—	8	1	no	—
168	D₂₄₄	—	2	2	no	—
169	D₂₄₄	C₂×C₄⋊C₄	1	4	yes	—
170	C₄×D₆₁	—	2	4	no	—
171	C₄×D₆₁	—	2	4	no	—
172	Dic₁₂₂	—	2	2	no	—
173	Dic₁₂₂	C₂×C₄⋊C₄	1	4	yes	—
174	C₈×C₆₁	—	2	4	no	—
175	C₈×C₆₁	—	2	2	no	—
176	C₈×C₆₁	—	4	2	no	—
177	C₈×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
178	C₈×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
179	C₈×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
180	C₈×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
181	#92418	—	2	2	no	—
182	#92418	—	4	2	no	—
183	#92418	C₄×D₄	1	2	yes	—
184	#92418	C₄×D₄	1	2	yes	—
185	#22695	—	61	1	no	maximal, Sylow
186	C₂²×C₆₁	—	4	1	no	—
187	C₂²×C₆₁	#520	1	1	yes	characteristic
188	C₂²×C₆₁	#547	1	1	yes	characteristic
189	C₂²×C₆₁	D₄×Q₈	1	1	yes	characteristic
190	C₂×D₆₁	—	2	4	no	—
191	C₂×D₆₁	—	8	2	no	—
192	C₄×C₆₁	—	8	1	no	—
193	C₄×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
194	C₄×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
195	C₄×C₆₁	D₄²	1	1	yes	characteristic
196	C₄×C₆₁	#554	1	1	yes	characteristic
197	Dic₆₁	—	2	4	no	—
198	Dic₆₁	—	4	2	no	—
199	Dic₆₁	—	8	1	no	—
200	#2957	—	61	1	no	—
201	#2957	—	61	1	no	—
202	C₂×#444	—	61	1	no	—
203	#2916	—	61	2	no	—
204	#1953	—	61	1	no	—
205	#1953	D₆₁	1	1	yes	characteristic, complemented
206	#1861	—	61	2	no	—
207	#1861	—	61	2	no	—
208	#1849	—	61	1	no	—
209	#1814	—	61	2	no	—
210	#1773	—	61	1	no	—
211	C₂×C₆₁	—	2	1	no	—
212	C₂×C₆₁	—	2	1	no	—
213	C₂×C₆₁	—	2	1	no	—
214	C₂×C₆₁	D₄×C₄⋊C₄	1	1	yes	characteristic
215	D₆₁	—	4	4	no	—
216	C₂×#135	—	61	1	no	—
217	#568	—	61	2	no	—
218	#568	—	61	2	no	—
219	C₂×#131	—	61	2	no	—
220	#444	—	61	4	no	—
221	#444	—	61	4	no	—
222	#444	—	122	2	no	—
223	#444	—	122	2	no	—
224	#443	—	122	2	no	—
225	#443	—	122	2	no	—
226	#430	—	61	2	no	—
227	#430	—	122	4	no	—
228	#430	C₂×D₆₁	1	2	yes	—
229	C₂×#108	—	61	1	no	—
230	C₂×#108	—	61	2	no	—
231	C₂×#108	—	61	1	no	—
232	C₂×#108	—	122	4	no	—
233	C₂×#108	C₂×D₆₁	1	1	yes	characteristic
234	C₂×#108	C₂×D₆₁	1	1	yes	characteristic
235	#427	—	61	1	no	—
236	#423	—	61	2	no	—
237	#421	—	61	2	no	—
238	#421	—	61	2	no	—
239	C₂×#104	—	61	2	no	—
240	#405	—	61	1	no	—
241	#405	C₂×D₆₁	1	1	yes	characteristic
242	C₂×#97	—	61	1	no	—
243	#344	—	61	2	no	—
244	#344	C₂×D₆₁	1	2	yes	—
245	#339	—	61	1	no	—
246	#339	—	61	1	no	—
247	C₆₁	#22695	1	1	yes	characteristic, complemented, Sylow
248	C₂×#41	—	61	2	no	—
249	#135	—	61	2	no	—
250	#135	—	61	2	no	—
251	#135	—	61	2	no	—
252	#135	—	61	2	no	—
253	C₂×Q₁₆	—	61	1	no	—
254	C₂×SD₁₆	—	61	2	no	—
255	C₂×D₈	—	61	1	no	—
256	#131	—	61	4	no	—
257	#131	—	61	4	no	—
258	#131	—	122	4	no	—
259	#131	—	122	2	no	—
260	#131	—	122	2	no	—
261	#131	—	122	4	no	—
262	#131	—	244	2	no	—
263	#131	—	244	2	no	—
264	C₂×M₁₆	—	61	1	no	—
265	C₂×M₁₆	—	61	2	no	—
266	C₂×M₁₆	—	122	2	no	—
267	C₂×M₁₆	—	122	4	no	—
268	C₂×M₁₆	—	122	4	no	—
269	C₂×M₁₆	C₂²×D₆₁	1	2	yes	—
270	C₂×M₁₆	C₂²×D₆₁	1	2	yes	—
271	C₂×M₁₆	C₂²×D₆₁	1	1	yes	characteristic
272	C₂²×C₈	—	61	1	no	—
273	C₂²×C₈	—	122	2	no	—
274	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
275	#117	—	61	1	no	—
276	#117	C₂²×D₆₁	1	1	yes	characteristic
277	C₂×C₄²	—	61	1	no	—
278	#108	—	2	4	no	—
279	#108	—	2	1	no	—
280	#108	—	2	1	no	—
281	#108	—	61	4	no	—
282	#108	—	122	4	no	—
283	#108	—	122	2	no	—
284	#108	—	122	2	no	—
285	#108	—	122	1	no	—
286	#108	—	122	1	no	—
287	#108	C₄×D₆₁	1	2	yes	—
288	#108	C₄×D₆₁	1	2	yes	—
289	#107	—	122	1	no	—
290	#106	—	122	1	no	—
291	C₄⋊C₈	—	244	2	no	—
292	#104	—	61	8	no	—
293	#104	—	122	4	no	—
294	#104	—	122	4	no	—
295	#97	—	2	1	no	—
296	#97	—	61	2	no	—
297	#97	—	61	2	no	—
298	#97	—	122	1	no	—
299	C₄×C₈	—	2	1	no	—
300	C₄×C₈	—	122	1	no	—
301	#41	—	61	4	no	—
302	#41	—	61	4	no	—
303	#41	—	122	2	no	—
304	#41	—	122	2	no	—
305	C₂×Q₈	—	61	2	no	—
306	C₂×D₄	—	61	2	no	—
307	C₂×D₄	—	122	2	no	—
308	C₂²×C₄	—	122	2	no	—
309	C₂²×C₄	—	122	1	no	—
310	C₂²×C₄	C₂³×D₆₁	1	1	yes	characteristic
311	Q₁₆	—	61	2	no	—
312	Q₁₆	—	61	2	no	—
313	SD₁₆	—	61	4	no	—
314	SD₁₆	—	61	4	no	—
315	D₈	—	61	2	no	—
316	D₈	—	61	2	no	—
317	M₁₆	—	2	2	no	—
318	M₁₆	—	2	1	no	—
319	M₁₆	—	2	2	no	—
320	M₁₆	—	2	2	no	—
321	M₁₆	—	61	2	no	—
322	M₁₆	—	122	4	no	—
323	M₁₆	—	122	4	no	—
324	M₁₆	—	122	1	no	—
325	M₁₆	—	122	2	no	—
326	M₁₆	—	122	2	no	—
327	M₁₆	—	244	4	no	—
328	M₁₆	—	244	4	no	—
329	M₁₆	C₂×C₄×D₆₁	1	4	yes	—
330	M₁₆	D₄×D₆₁	1	2	yes	—
331	C₂×C₈	—	2	1	no	—
332	C₂×C₈	—	2	2	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	61	2	no	—
335	C₂×C₈	—	61	2	no	—
336	C₂×C₈	—	61	1	no	—
337	C₂×C₈	—	61	1	no	—
338	C₂×C₈	—	122	4	no	—
339	C₂×C₈	—	122	4	no	—
340	C₂×C₈	—	122	1	no	—
341	C₂×C₈	—	122	2	no	—
342	C₂×C₈	—	244	4	no	—
343	C₂×C₈	—	244	4	no	—
344	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
345	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
346	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
347	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
348	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
349	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
350	C₄⋊C₄	—	2	1	no	—
351	C₄⋊C₄	—	122	1	no	—
352	C₂²⋊C₄	—	2	1	no	—
353	C₂²⋊C₄	—	122	1	no	—
354	C₄²	—	2	1	no	—
355	C₄²	—	61	2	no	—
356	C₄²	—	122	1	no	—
357	C₄²	—	122	1	no	—
358	C₂³	—	122	2	no	—
359	C₂³	#14519825	1	1	yes	characteristic
360	Q₈	—	61	4	no	—
361	Q₈	—	122	2	no	—
362	D₄	—	61	4	no	—
363	D₄	—	122	4	no	—
364	D₄	—	122	4	no	—
365	D₄	—	122	2	no	—
366	C₂×C₄	—	4	1	no	—
367	C₂×C₄	—	122	2	no	—
368	C₂×C₄	—	122	4	no	—
369	C₂×C₄	—	122	4	no	—
370	C₂×C₄	—	122	2	no	—
371	C₂×C₄	—	488	1	no	—
372	C₂×C₄	—	244	1	no	—
373	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
374	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
375	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
376	C₂×C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
377	C₂×C₄	#41×D₆₁	1	1	yes	characteristic
378	C₂×C₄	#14519823	1	1	yes	characteristic
379	C₈	—	2	4	no	—
380	C₈	—	2	2	no	—
381	C₈	—	4	2	no	—
382	C₈	—	61	2	no	—
383	C₈	—	61	2	no	—
384	C₈	—	122	2	no	—
385	C₈	—	244	2	no	—
386	C₈	C₄⋊C₄×D₆₁	1	1	yes	characteristic
387	C₈	C₄⋊C₄×D₆₁	1	1	yes	characteristic
388	C₈	C₄⋊C₄×D₆₁	1	1	yes	characteristic
389	C₈	C₄⋊C₄×D₆₁	1	1	yes	characteristic
390	C₂²	—	4	1	no	—
391	C₂²	—	122	4	no	—
392	C₂²	—	488	2	no	—
393	C₂²	#16840244	1	1	yes	characteristic
394	C₂²	#16840414	1	1	yes	characteristic
395	C₂²	#16840427	1	1	yes	characteristic
396	C₄	—	8	1	no	—
397	C₄	—	122	4	no	—
398	C₄	—	488	1	no	—
399	C₄	—	244	2	no	—
400	C₄	C₂×C₄⋊C₄×D₆₁	1	1	yes	characteristic
401	C₄	C₄×D₄×D₆₁	1	1	yes	characteristic
402	C₄	#16840396	1	1	yes	characteristic
403	C₄	#16840468	1	1	yes	characteristic
404	C₂	—	2	1	no	—
405	C₂	—	2	1	no	—
406	C₂	—	2	1	no	—
407	C₂	—	244	4	no	—
408	C₂	#21448720	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16840396
commutator	C₂×C₄×C₆₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₁
Fitting	C₆₁×#1953	C₂
socle	C₂×C₆₁	D₄×C₄⋊C₄
prime radical	C₂×#14519799	C₂²
cosocle	C₂²×C₄×C₆₁	C₂⁴
hypercenter	#1953	D₆₁
hypocenter	C₆₁	#22695
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1953

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
4. #21454429	C₂	semidirect
5. #21446610	C₂	semidirect
6. #21446062	C₂	semidirect
7. #21446056	C₂	semidirect
8. #21445992	C₂	semidirect
9. #21445872	C₂	semidirect
10. #21445704	C₂	semidirect
11. C₆₁×#1953	C₂	semidirect
18. #16839883	C₄	semidirect
19. #16839883	C₄	semidirect
205. #1953	D₆₁	semidirect
247. C₆₁	#22695	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#21454737	2
#21454735	2
C₂×#16839883	2
#21454429	2
#21446610	2
#21446062	2
#21446056	2
#21445992	2
#21445872	2
#21445704	2
C₆₁×#1953	2
#22695	61